Corollary 13.7.9D73416504271

Proof. Lower inequality. Let $B$ be a coordinate ball such that $\overline B$ lies in a larger coordinate ball, add fixed smooth local potentials of the $\theta _ j$, defined near $\overline B$, and denote the resulting plurisubharmonic functions on $B$ by $v_{j,k}$ and $v_ j$. By lemma 13.7.3, $v_{j,k}\to v_ j$ in extended capacity on $B$, and the $v_{j,k}$ are uniformly bounded above on $B$, because the suprema of the $u_{j,k}$ are bounded above; the local products are finite by theorem 13.5.4. For $C,\varepsilon >0$ let $v_{j,k}^ C$, $v_ j^ C$ and $a_ k$, $a$ be the truncations and cutoff weights of lemma 13.7.4; thus $0\leq a_ k,a\leq 1$, $a$ is quasi-continuous and $a_ k\to a$ locally in capacity. We may assume $A>0$. The bounded Borel weights $\chi _ ka_ k$ converge locally in capacity to the bounded quasi-continuous function $\chi a$, because $|\chi _ ka_ k-\chi a|\leq A|a_ k-a|+|\chi _ k-\chi |$, so that the set where $|\chi _ ka_ k-\chi a|>\delta $ is contained in the union of the sets where $|a_ k-a|>\delta /(2A)$ and where $|\chi _ k-\chi |>\delta /2$. The product $\chi a$ is quasi-continuous, since $\chi $ and $a$ are continuous outside open sets of arbitrarily small capacity and capacity is subadditive.

By lemma 13.7.4(2) and $0\leq a_ k\leq 1$,

The functions $v_{j,k}^ C\geq -C$ are uniformly bounded on $B$ and converge to $v_ j^ C$ locally in capacity (lemma 13.7.3). By the weighted capacity continuity of bounded products (theorem 7.5.8), the right side converges weakly to $\chi a\bigwedge _ j\mathrm{dd}^{\mathrm{c}}v_ j^ C=\chi aR$, the last identity by lemma 13.7.4(2). Testing against $g\eta $ for a nonnegative test function $g$ on $B$, we obtain $\int g\chi aR\wedge \eta \leq \varliminf _ k\int g\chi _ kR_ k\wedge \eta $, and we let $\varepsilon \downarrow 0$ and then $C\to \infty $. By lemma 13.7.4(3), the weight $a$ increases to one outside the union of the pole sets, which $R$ does not charge, so monotone convergence gives the local lower inequality, and a finite partition of unity (proposition 5.2.6) gives the global one. The local argument uses only the local upper bounds, the extended capacity convergence and the finiteness of the local products.

Convergence. Under the additional mass hypothesis, theorem 13.7.5 gives $R_ k\to R$ weakly, and in particular $\int _ XR_ k\wedge \eta \to \int _ XR\wedge \eta $. Apply the lower inequality also to the weights $A-\chi _ k$, which converge in capacity to $A-\chi $. Since

this gives the matching upper limit, so $\int _ X\chi _ kR_ k\wedge \eta \to \int _ X\chi R\wedge \eta $.

Every smooth real $(n-p,n-p)$-form $\eta $ on $X$ is the difference of two smooth strongly positive forms. Indeed, if $(\rho _ i)$ is a finite partition of unity subordinate to coordinate charts (proposition 5.2.6), then $\rho _ i\eta =\sum _ lf_ le_ l$ with smooth real functions $f_ l$ supported in $\operatorname{Supp}\rho _ i$ and constant strongly positive forms $e_ l$, which span the real forms by proposition 5.3.2. If $\kappa _ i\geq 0$ is a test function of the chart equal to one on $\operatorname{Supp}\rho _ i$ and $c_ l\geq \sup |f_ l|$, then $f_ le_ l=(f_ l+c_ l\kappa _ i)e_ l-c_ l\kappa _ ie_ l$ is a difference of smooth strongly positive forms. Hence the convergence holds for all smooth real forms, and by linearity for all smooth forms, so $\chi _ kR_ k\to \chi R$ weakly as currents. Since $\chi _ k\leq A$ and the trace masses of $R_ k$ are bounded, the coefficient measures of $\chi _ kR_ k$ have bounded total variation; when $p=n$, approximating continuous functions on the compact manifold $X$ uniformly by smooth ones, using a partition of unity and lemma 2.5.1, upgrades the convergence to weak convergence of Radon measures. Finally, for monotone sequences the extended capacity hypothesis follows from proposition 13.7.8, applied to the local potentials. $\square$